A287825 -id:A287825 - cp's OEIS frontend

A287834 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 3.

1, 11, 105, 1005, 9621, 92105, 881753, 8441329, 80811789, 773639469, 7406320733, 70903294113, 678781988705, 6498216958121, 62209699634757, 595555173609653, 5701457600593525, 54582044135967257, 522532964509030377, 5002390498942001761, 47889630709552579709

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,-2,-21,10).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, -2, -21, 10}, {1, 11, 105, 1005, 9621}, 20]

def a(n):
 if n in [0,1,2,3,4]:
  return [1, 11, 105, 1005, 9621][n]
 return 10*a(n-1) - 2*a(n-2) - 21*a(n-3) + 10*a(n-4)

a(n) = 10*a(n-1) - 2*a(n-2) - 21*a(n-3) + 10*a(n-4), a(0)=1, a(1)=11, a(2)=105, a(3)=1005, a(4)=9621.

G.f.: (-1 - x + 3*x^2 + 2*x^3 - 2*x^4)/(-1 + 10*x - 2*x^2 - 21*x^3 + 10*x^4).

A287835 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 4.

Original entry on oeis.org

1, 11, 107, 1043, 10169, 99149, 966719, 9425675, 91901945, 896059709, 8736735695, 85184670011, 830565128489, 8098152315149, 78958372642847, 769857662314475, 7506244118089817, 73187166301583837, 713587411625345903, 6957599532298617755, 67837787583138657929

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,-1,-14)

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, -1, -14}, {1, 11, 107, 1043}, 20]

def a(n):
 if n in [0,1,2,3]:
  return [1, 11, 107, 1043][n]
 return 10*a(n-1) - a(n-2) - 14*a(n-3)

For n>3, a(n) = 10*a(n-1) - a(n-2) - 14*a(n-3), a(0)=1, a(1)=11, a(2)=107, a(3)=1043.

G.f.: (1 + x - 2 x^2 - 2 x^3)/(1 - 10 x + x^2 + 14 x^3).

A287836 Number of words over the alphabet {0,1,...,10} such that no two consecutive terms have distance 5.

Original entry on oeis.org

1, 11, 109, 1081, 10721, 106329, 1054553, 10458881, 103729441, 1028771337, 10203182953, 101193470929, 1003620008177, 9953736259545, 98719500126905, 979083577381409, 9710388021269185, 96306012787788969, 955147011506293513, 9472989143467878769, 93951530216004879761

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,1,-18).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, 1, -18}, {1, 11, 109, 1081}, 20]

def a(n):
 if n in [0,1,2,3]:
  return [1, 11, 109, 1081][n]
 return 10*a(n-1) + a(n-2) - 18*a(n-3)

For n>3, a(n) = 10*a(n-1) + a(n-2) - 18*a(n-3), a(0)=1, a(1)=11, a(2)=109, a(3)=1081.

G.f.: (1 + x - 2*x^2 - 2*x^3)/(1 - 10*x - x^2 + 18*x^3).

A287837 Number of words over the alphabet {0,1,...,10} such that no two consecutive terms have distance 7.

Original entry on oeis.org

1, 11, 113, 1163, 11969, 123179, 1267697, 13046507, 134268161, 1381821131, 14221015793, 146355621323, 1506219260609, 15501259470059, 159531252482417, 1641816303234347, 16896756789790721, 173893016807610251, 1789620438445474673, 18417883434877577483

Offset: 0

David Nacin, Jun 07 2017

In general, the number of sequences on {0,1,...,10} such that no two consecutive terms have distance 6+k for k in {0,1,2,3,4} has generating function (-1 - x)/(-1 + 10*x + (2*k+1)*x^2).

Index entries for linear recurrences with constant coefficients, signature (10,3).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, 3}, {1, 11, 113}, 20]

def a(n):
 if n in [0,1,2]:
  return [1, 11, 113][n]
 return 10*a(n-1) + 3*a(n-2)

For n>2, a(n) = 10*a(n-1) + 3*a(n-2), a(0)=1, a(1)=11, a(2)=113.
G.f.: (-1 - x)/(-1 + 10*x + 3*x^2).
a(n) = A015588(n)+A015588(n+1). - R. J. Mathar, Oct 20 2019

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A287834 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 3.

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A287835 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 4.

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A287836 Number of words over the alphabet {0,1,...,10} such that no two consecutive terms have distance 5.

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A287837 Number of words over the alphabet {0,1,...,10} such that no two consecutive terms have distance 7.

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